## Cartan matrices and Dynkin diagrams

The Cartan matrix is $C= ⟨ αi∨ , αi ⟩ with αi∨ = 2αi ⟨ αi , αi ⟩$ So that the matrix of the form is $A= ⟨ αi , αj ⟩ = DC where D=diag ⟨ αi , αi ⟩ 2 .$

Type ${A}_{n-1}$: The Dynkin diagram is

and $C= 2 -1 -1 2 -1 -1 2 -1 ⋱ 0 0 2 -1 -1 2 =A.$

Type ${A}_{5}$: The Dynkin diagram is

and

$Q= 0 v-u 1 1 1 u-v 0 v-u 1 1 1 u-v 0 v-u 1 1 1 u-v 0 v-u 1 1 1 u-v 0$

Type ${B}_{n}$: The Dynkin diagram is

and $C= 2 -2 -1 2 -1 -1 2 -1 ⋱ 0 0 2 -1 -1 2 and A= 2 -2 -2 4 -2 -2 4 -2 ⋱ 0 0 4 -2 -2 4 ,$ and $D=diag\left(1,2,2,\dots ,2\right)$.

Type ${B}_{5}$: The Dynkin diagram is

and

$Q= 0 v-u2 1 1 1 u-v2 0 v-u 1 1 1 u-v 0 v-u 1 1 1 u-v 0 v-u 1 1 1 u-v 0$

Type ${C}_{n}$: The Dynkin diagram is

and $C= 2 -1 -2 2 -1 -1 2 -1 ⋱ 0 0 2 -1 -1 2 and A= 4 -2 -2 2 -1 -1 2 -1 ⋱ 0 0 2 -1 -1 2 ,$ and $D=diag\left(2,1,1,\dots ,1\right)$.

Type ${C}_{5}$: The Dynkin diagram is

and

$Q= 0 v2-u 1 1 1 u2-v 0 v-u 1 1 1 u-v 0 v-u 1 1 1 u-v 0 v-u 1 1 1 u-v 0$

Type ${D}_{n}$: The Dynkin diagram is

and $C= 2 0 -1 0 2 -1 -1 -1 2 -1 -1 2 -1 ⋱ 0 0 2 -1 -1 2 =A.$

Type ${D}_{5}$: The Dynkin diagram is

and

$Q= 0 1 v-u 1 1 1 0 v-u 1 1 u-v u-v 0 v-u 1 1 1 u-v 0 v-u 1 1 1 u-v 0$

Type ${E}_{6}$: The Dynkin diagram is

and $C= 2 0 -1 0 0 0 0 2 0 -1 0 0 -1 0 2 -1 0 0 0 -1 -1 2 -1 0 0 0 0 -1 2 -1 0 0 0 0 -1 2 =A and Q= 0 1 v-u 1 1 1 1 0 1 v-u 1 1 u-v 1 0 v-u 1 1 1 u-v u-v 0 v-u 1 1 1 1 u-v 0 v-u 1 1 1 1 u-v 0 .$

Type ${E}_{7}$: The Dynkin diagram is

and $C= 2 0 -1 0 0 0 0 0 2 0 -1 0 0 0 -1 0 2 -1 0 0 0 0 -1 -1 2 -1 0 0 0 0 0 -1 2 -1 0 0 0 0 0 -1 2 -1 0 0 0 0 0 -1 2 =A and Q= 0 1 v-u 1 1 1 1 1 0 1 v-u 1 1 1 u-v 1 0 v-u 1 1 1 1 u-v u-v 0 v-u 1 1 1 1 1 u-v 0 v-u 1 1 1 1 1 u-v 0 v-u 1 1 1 1 1 u-v 0 .$

Type ${E}_{8}$:

and $C= 2 0 -1 0 0 0 0 0 0 2 0 -1 0 0 0 0 -1 0 2 -1 0 0 0 0 0 -1 -1 2 -1 0 0 0 0 0 0 -1 2 -1 0 0 0 0 0 0 -1 2 -1 0 0 0 0 0 0 -1 2 -1 0 0 0 0 0 0 -1 2 =A and Q= 0 1 v-u 1 1 1 1 1 1 0 1 v-u 1 1 1 1 u-v 1 0 v-u 1 1 1 1 1 u-v u-v 0 v-u 1 1 1 1 1 1 u-v 0 v-u 1 1 1 1 1 1 u-v 0 v-u 1 1 1 1 1 1 u-v 0 v-u 1 1 1 1 1 1 u-v 0 .$

Type ${F}_{4}$: The Dynkin diagram is

and $C= 2 -1 0 0 -1 2 -2 0 0 -1 2 -1 0 0 -1 2 and A= 2 -1 0 0 -1 2 -2 0 0 -2 4 -2 0 0 -2 4 and Q= 0 v-u 1 1 u-v 0 v-u2 1 1 u-v2 0 v-u 1 1 u-v 0$ and $D=diag\left(1,1,2,2\right)$.

Type ${G}_{2}$: The Dynkin diagram is

and $C= 2 -3 -1 2 , A= 2 -3 -3 6 , Q= 0 v-u3 u-v3 0 , with 1 0 0 3 .$ and $D=diag\left(1,1,2,2\right)$.

Type ${A}_{5}^{\left(1\right)}$:The Dynkin diagram is

and

$Q= 0 v-u 1 1 u-v u-v 0 v-u 1 1 1 u-v 0 v-u 1 1 1 u-v 0 v-u v-u 1 1 u-v 0$

Passing to the nonsimply laced case (following Reinecke and Lusztig's book), $C˜ = a ˜ ij i,j∈ I ˜ an indecomposable Cartan matrix.$ $\stackrel{˜}{C}$ is a quotient of a simply laced Cartan matrix $C={\left({a}_{ij}\right)}_{i,j\in I}$ under the action of a cyclic group $Z$ on $I$. Then There is an induced action of $Z$ on ${U}^{+}$ and on $B\left(\infty \right).$ Then In this form ${C}_{n}$ comes from ${A}_{2n-1}$:

 $→ ℤ/2ℤ=Z$
${B}_{n}$ comes from ${D}_{n+1}$:
 $→ ℤ/2ℤ=Z$
${F}_{4}$ comes from ${E}_{6}$:
 $→ ℤ/2ℤ=Z$
${G}_{2}$ comes from ${D}_{4}$:
 $→ ℤ/3ℤ=Z$
${A}_{l-1}^{\left(1\right)}$ comes from ${A}_{\infty }$:
 $→ ℤℤ=Z$

## References

[D] V.G. Drinfeld, Quantum Groups, Vol. 1 of Proccedings of the International Congress of Mathematicians (Berkeley, Calif., 1986). Amer. Math. Soc., Providence, RI, 1987, pp. 198–820. MR0934283

[DHL] H.-D. Doebner, Hennig, J. D. and W. Lücke, Mathematical guide to quantum groups, Quantum groups (Clausthal, 1989), Lecture Notes in Phys., 370, Springer, Berlin, 1990, pp. 29–63. MR1201823

[J] N. Jacobson, Lie algebras, Interscience Publishers, New York, 1962.